matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs.ma

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  14
  15 include "basic_2/notation/relations/relationstarstar_4.ma".
  16 include "basic_2/static/lfxs.ma".
  17
  18 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
  19
  20 definition tc_lfxs (R) (T): relation lenv ≝ TC … (lfxs R T).
  21
  22 interpretation "iterated extension on referred entries (local environment)"
  23    'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
  24
  25 (* Basic properties *********************************************************)
  26
  27 lemma tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
  28 /2 width=1 by inj/ qed.
  29
  30 lemma tc_lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
  31                     L1 ⦻**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
  32 #R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
  33 /3 width=4 by lfxs_sort, step, inj/
  34 qed.
  35
  36 lemma tc_lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
  37                     L1 ⦻**[R, #i] L2 → L1.ⓑ{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
  38 #R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
  39 /3 width=4 by lfxs_lref, step, inj/
  40 qed.
  41
  42 lemma tc_lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
  43                     L1 ⦻**[R, §l] L2 → L1.ⓑ{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
  44 #R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
  45 /3 width=4 by lfxs_gref, step, inj/
  46 qed.
  47
  48 lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
  49                    (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
  50                    ∀T. symmetric … (tc_lfxs R T).
  51 #R #H1R #H2R #T #L1 #L2 #H elim H -L2
  52 /4 width=3 by lfxs_sym, TC_strap, inj/
  53 qed-.
  54
  55 lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
  56                   ∀L1,L2,T. L1 ⦻**[R1, T] L2 → L1 ⦻**[R2, T] L2.
  57 #R1 #R2 #HR #L1 #L2 #T #H elim H -L2
  58 /4 width=5 by lfxs_co, step, inj/
  59 qed-.
  60
  61 (* Basic inversion lemmas ***************************************************)
  62
  63 lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆.
  64 #R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
  65 qed-.
  66
  67 lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆.
  68 #R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
  69 /3 width=3 by inj, lfxs_inv_atom_dx/
  70 qed-.
  71
  72 lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻**[R, ⋆s] Y2 →
  73                         (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  74                         ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 &
  75                                          Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  76 #R #Y1 #Y2 #s #H elim H -Y2
  77 [ #Y2 #H elim (lfxs_inv_sort … H) -H *
  78   /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
  79 | #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
  80   [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
  81   | #I #L #L2 #V #V2 #HL2 #H #H2 * *
  82     [ #H1 #H0 destruct
  83     | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
  84       /4 width=8 by ex3_5_intro, step, or_intror/
  85     ]
  86   ]
  87 ]
  88 qed-.
  89 (*
  90 lemma tc_lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻**[R, #0] Y2 →
  91                         (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  92                         ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, V1] L2 & R L1 V1 V2 &
  93                                          Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  94 #R #Y1 #Y2 #H elim H -Y2
  95 [ #Y2 #H elim (lfxs_inv_zero … H) -H *
  96   /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/
  97 | #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H *
  98   [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
  99   | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * *
 100     [ #H1 #H0 destruct
 101     | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct
 102       @or_intror @ex4_5_intro [6,7: |*: /width=7/ ]
 103
 104       /4 width=8 by ex3_5_intro, step, or_intror/
 105     ]
 106   ]
 107 ]
 108 qed-.
 109
 110
 111
 112
 113
 114 #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
 115 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
 116 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
 117   /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
 118 ]
 119 qed-.
 120
 121 lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
 122                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 123                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
 124                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 125 #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
 126 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
 127 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
 128   /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
 129 ]
 130 qed-.
 131
 132 lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
 133                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 134                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
 135                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 136 #R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
 137 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
 138 | lapply (frees_inv_gref … H1) -H1 #Hf
 139   elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
 140   elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
 141   /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
 142 ]
 143 qed-.
 144
 145 lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
 146                      L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 147 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
 148 /6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 149 qed-.
 150
 151 lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
 152                      L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
 153 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
 154 /5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 155 qed-.
 156
 157 (* Advanced inversion lemmas ************************************************)
 158
 159 lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
 160                              ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
 161 #R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
 162 [ #H destruct
 163 | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 164 ]
 165 qed-.
 166
 167 lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
 168                              ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
 169 #R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
 170 [ #_ #H destruct
 171 | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 172 ]
 173 qed-.
 174
 175 lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
 176                              ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 177                                       Y2 = L2.ⓑ{I}V2.
 178 #R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
 179 [ #H destruct
 180 | #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
 181   /2 width=5 by ex3_2_intro/
 182 ]
 183 qed-.
 184
 185 lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
 186                              ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 187                                       Y1 = L1.ⓑ{I}V1.
 188 #R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
 189 [ #_ #H destruct
 190 | #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
 191   /2 width=5 by ex3_2_intro/
 192 ]
 193 qed-.
 194
 195 lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
 196                              ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
 197 #R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
 198 [ #H destruct
 199 | #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 200 ]
 201 qed-.
 202
 203 lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
 204                              ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
 205 #R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
 206 [ #_ #H destruct
 207 | #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 208 ]
 209 qed-.
 210
 211 lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
 212                              ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
 213 #R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
 214 [ #H destruct
 215 | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 216 ]
 217 qed-.
 218
 219 lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
 220                              ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
 221 #R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
 222 [ #_ #H destruct
 223 | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 224 ]
 225 qed-.
 226
 227 (* Basic forward lemmas *****************************************************)
 228
 229 lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
 230 #R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
 231 /4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
 232 qed-.
 233
 234 lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
 235                         R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 236 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
 237 qed-.
 238
 239 lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
 240 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 241 qed-.
 242
 243 lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
 244 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 245 qed-.
 246
 247 lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
 248 #R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
 249 qed-.
 250
 251 (* Basic_2A1: removed theorems 24:
 252               llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
 253               llpx_sn_bind llpx_sn_flat
 254               llpx_sn_inv_bind llpx_sn_inv_flat
 255               llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
 256               llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
 257               llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
 258               llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
 259 *)
 260 *)